MathDB

4

Part of 1993 Romania Team Selection Test

Problems(3)

Integral solutions

Source: Romania TST 1993

7/12/2009
Prove that the equation (x\plus{}y)^n\equal{}x^m\plus{}y^m has a unique solution in integers with x>y>0 x>y>0 and m,n>1 m,n>1.
number theorydiophantine
f(A) = f(B) = max A\triangle B

Source: Romania IMO TST 1993 1.4

2/17/2020
Let YY be the family of all subsets of X={1,2,...,n}X = \{1,2,...,n\} (n>1n > 1) and let f:Y→Xf : Y \to X be an arbitrary mapping. Prove that there exist distinct subsets A,BA,B of XX such that f(A)=f(B)=maxA△Bf(A) = f(B) = max A\triangle B, where A△B=(A−B)∪(B−A)A\triangle B = (A-B)\cup(B-A).
Subsetsfunctionalgebra
max of \frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+...}

Source: Romania IMO TST 1993 2.4

2/17/2020
For each integer n>3n > 3 find all quadruples (n1,n2,n3,n4)(n_1,n_2,n_3,n_4) of positive integers with n1+n2+n3+n4=nn_1 +n_2 +n_3 +n_4 = n which maximize the expression n!n1!n2!n3!n4!2(n12)+(n22)+(n32)+(n42)+n1n2+n2n3+n3n4\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}
combinatoricsCombinationsBinomialmaxinequalities